Asymptotic expansion of the voltage potential under small Robin perturbations of its boundary conditions

Abstract

Inspired by questions related to inverse problems and shape optimization, we derive an asymptotic expansion of the voltage potential, solution to a model elliptic second-order partial differential equation, under small perturbations of its boundary conditions. More precisely, the homogeneous Dirichlet or homogeneous Neumann boundary condition in a fixed, reference configuration of the problem is replaced by a Robin boundary condition with admittance $k_\varepsilon > 0$ on a ``small'' subset $ω_\varepsilon$ of the boundary of the ambient domain, vanishing at the limit $\varepsilon \to 0$. In each of these two situations, a general asymptotic formula is established for the voltage potential, which rests on minimal assumptions about the shape of the vanishing subset $ω_\varepsilon$ and the parameter $k_\varepsilon$. The scalings of these expansions, capturing the intensity of the perturbation, are measured by new quantities called ``Robin-Dirichlet'' capacity or a ``Robin-Neumann'' capacity, which depend on the geometry of $ω_\varepsilon$ and on the value of the parameter $k_\varepsilon$. We analyze how these quantities compare to more classical measures of the ``smallness'' of $ω_\varepsilon$, such as the capacity or the Neumann capacity of $ω_\varepsilon$, according to the behavior of $k_\varepsilon$ as $\varepsilon \to 0$.